In the general case, the situation is somewhat more complicated.
Let \(\varphi \in \Phi _w(\Omega )\) .
By [1]}, if \(\Vert u\Vert _\varphi < 1\) , then \(\varrho _\varphi (u) \lesssim \Vert  u\Vert _\varphi \) .
Thus \(\operatorname{M_{\varphi }}(\Gamma ) = 0\)  implies \(\operatorname{\widetilde{M}_{\varphi }}(\Gamma ) = 0\) .
The converse implication does not necessarily hold, as the next example shows, which is the main reason for using norms instead of modulars in Definition REF .
